latex/schrodinger_simulation.tex

@@ -99,7 +99,7 @@
 \subfile{sections/methods}
 
 % Results
-% \subfile{sections/results}
+\subfile{sections/results}
 
 % Conclusion
 % \subfile{sections/conclusion}

latex/sections/appendices.tex

@@ -34,15 +34,64 @@ Using Equation \eqref{eq:schrodinger_dimensionless}, we get
     &= -\frac{i \Delta t}{2} \bigg[ - \frac{u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1}}{2 \Delta x^{2}} \\
     & \quad - \frac{u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1}}{2 \Delta y^{2}} + \frac{1}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\
     & \quad - \frac{u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n}}{2 \Delta x^{2}} \\
-    & \quad - \frac{u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n}}{2 \Delta y^{2}} + \frac{1}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \bigg] \\
+    & \quad - \frac{u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n}}{2 \Delta y^{2}} + \frac{1}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \bigg]
 \end{align*}
-We rewrite the expression, 
+We rewrite the expression and gather all terms containing the $n+1$ time step on 
+the left hand side, and the terms containing $n$ time step on the right hand side. 
 \begin{align*}
     & u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\
     & - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\
     &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\
-    & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \\
+    & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} 
+\end{align*}
+In addition, since we will use an equal step size $h$ in both $x$ and $y$ direction, 
+we can use 
+\begin{align*}
+    \frac{i \Delta t}{2 \Delta h^{2}} = \frac{i \Delta t}{2 \Delta x^{2}} = \frac{i \Delta t}{2 \Delta y^{2}} \ ,
+\end{align*}
+and define 
+\begin{align*}
+    r \equiv \frac{i \Delta t}{2 \Delta h^{2}}
+\end{align*}
+Now, the discretized Schrödinger equation can be written as 
+\begin{align*}
+    & u_{\ivec, \jvec}^{n+1} - r \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\
+    & - r \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\
+    &= u_{\ivec, \jvec}^{n} + r \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\
+    & \quad + r \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \ .
 \end{align*}
 
 
+\section{Matrix structure}\label{ap:matrix_structure}
+For $u$ vector of length $(M-2) = 3$, the matrices $A$ and $B$ have size 
+$(M-2)^{2} \times (M-2)^{2} = 9 \times 9$ given by 
+\begin{align*}
+    A = 
+    \begin{bmatrix}
+    a_{0} &  -r  &  0   &  -r   &   0   &  0    &  0   &  0    &  0    \\
+    -r  &  a_{1} &  -r  &  0    &   -r  &  0    &  0   &  0    &  0    \\
+    0   &  -r  &  a_{2} &  0    &   0   &  -r   &  0   &  0    &  0    \\
+    -r  &  0   &  0   &  a_{3}  &   -r  &  0    &  -r  &  0    &  0    \\
+    0   &  -r  &  0   &  -r   &   a_{4} &  -r   &  0   &  -r   &  0    \\
+    0   &  0   &  -r  &  0    &   -r  &  a_{5}  &  0   &  0    &  -r   \\
+    0   &  0   &  0   &  -r   &   0   &  0    &  a_{6} &  -r   &  0    \\
+    0   &  0   &  0   &  0    &   -r  &  0    &  -r  &  a_{7}  &  -r   \\
+    0   &  0   &  0   &  0    &   0   &  -r   &  0   &  -r   &  a_{8}  \\
+    \end{bmatrix}
+\end{align*}
+
+\begin{align*}
+    B = 
+    \begin{bmatrix}
+    b_{0} &  r   &  0   &  r    &   0   &  0    &  0   &  0    &  0    \\
+    r   &  b_{1} &  r   &  0    &   r   &  0    &  0   &  0    &  0    \\
+    0   &  r   &  b_{2} &  0    &   0   &  r    &  0   &  0    &  0    \\
+    r   &  0   &  0   &  b_{3}  &   r   &  0    &  r   &  0    &  0    \\
+    0   &  r   &  0   &  r    &   b_{4} &  r    &  0   &  r    &  0    \\
+    0   &  0   &  r   &  0    &   r   &  b_{5}  &  0   &  0    &  r    \\
+    0   &  0   &  0   &  r    &   0   &  0    &  b_{6} &  r    &  0    \\
+    0   &  0   &  0   &  0    &   r   &  0    &  r   &  b_{7}  &  r    \\
+    0   &  0   &  0   &  0    &   0   &  r    &  0   &  r    &  b_{8}  \\
+    \end{bmatrix}
+\end{align*}
 \end{document}

latex/sections/methods.tex

@@ -68,11 +68,15 @@ We get the Crank-Nicolson method (CN) when $\theta = 1/2$ gives Crank-Nicolson
 \end{align} %
 Using CN, we derive the discretized Schrödinger equation given by 
 \begin{align*}
-    & u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\
-    & - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\
-    &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\
-    & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \numberthis \ .
-    \label{eq:schrodinger_discretized} 
+    & u_{\ivec, \jvec}^{n+1} - r \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\
+    & - r \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\
+    &= u_{\ivec, \jvec}^{n} + r \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\
+    & \quad + r \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \numberthis \ ,
+    \label{eq:schrodinger_discretized}
+\end{align*} %
+where $r$ is defined as
+\begin{align*}
+    r \equiv \frac{i \Delta t}{2 \Delta h^{2}}
 \end{align*} %
 The full derivation of both Equation \eqref{eq:crank_nicolson_method} and Equation \eqref{eq:schrodinger_discretized} 
 can be found in Appendix \ref{ap:crank_nicolson}.
@@ -97,8 +101,20 @@ A, B are sparse csc matrix
 %       & \bullet & \bullet &  &  & \bullet &  &  & 
 % \end{pNiceArray}
 % \end{equation*}
-
-\begin{equation*}
+We use Dirichlet boundary conditions, as given in Table \ref{tab:boundary_conditions}, 
+which allows us to express Equation \eqref{eq:schrodinger_discretized} as a matrix 
+equation 
+\begin{align}
+    A u^{n+1} = B u^{n} \ .
+\end{align}
+Here, both $u^{n+1}$ and $u^{n}$ are column vectors containing the internal points 
+of the $xy$ grid at time step $n+1$ and $n$, respectively. Since we have $M$ points 
+in $x$- and $y$-direction, we have $M-2$ internal points. Both $u$ vectors have 
+length $(M-2)^{2}$, and the matrices $A$ and $B$ have size $(M-2)^{2} \times (M-2)^{2}$. 
+The matrices can be decomposed as submatrices of size $(M-2) \times (M-2)$, with 
+the following pattern 
+\begin{align*}
+    A, B = 
     \begin{bmatrix}
         \begin{matrix}
             \bullet & \bullet & \phantom{\bullet} \\
@@ -154,7 +170,13 @@ A, B are sparse csc matrix
            \phantom{\bullet} & \bullet & \bullet 
        \end{matrix}
     \end{bmatrix}
-\end{equation*}
+\end{align*}
+To fill the matrices $A$ and $B$, we used 
+\begin{align*}
+    a_{k} &= 1 + 4r + \frac{i \Delta t}{2} v_{\ivec, \jvec} \\
+    b_{k} &= 1 - 4r - \frac{i \Delta t}{2} v_{\ivec, \jvec} \ .
+\end{align*} 
+An example of filled matrices can be found in Appendix \ref{ap:matrix_structure}.
 
 Notations:
 In addition, we use an equal step size in x- and y-direction, $h$ such that 
@@ -172,7 +194,7 @@ which gives a matrix $U^{n}$ that contains elements $u_{\ivec, \jvec}^{n}$, and
 a matrix $V$ that contains elements $v_{\ivec, \jvec}$.
 \begin{table}[H]
     \centering
-    \begin{tabular}{l l l} % @{\extracolsep{\fill}}
+    \begin{tabular}{l r} % @{\extracolsep{\fill}}
         \hline
         Position & Value  \\
         \hline 
@@ -189,7 +211,7 @@ a matrix $V$ that contains elements $v_{\ivec, \jvec}$.
 For the general setup of the wall, we used 
 \begin{table}[H]
     \centering
-    \begin{tabular}{l l} % @{\extracolsep{\fill}}
+    \begin{tabular}{l r} % @{\extracolsep{\fill}}
         \hline
         Parameter & Value  \\
         \hline 
@@ -205,7 +227,7 @@ For the general setup of the wall, we used
 
 \begin{table}[H]
     \centering
-    \begin{tabular}{l l l} % @{\extracolsep{\fill}}
+    \begin{tabular}{l r r} % @{\extracolsep{\fill}}
         \hline
         Simulation & $1$ & $2$ \\
         \hline 

latex/sections/results.tex

@@ -4,8 +4,25 @@
 \section{Results}\label{sec:results}
 \subsection{Deviation}\label{ssec:deviation}
 % Problem 3: Discuss approaches to solve Au^{n+1} = b, dealing with sparse matrix...
+We used the superlu solver, which is a dedicated solver for sparse matrices. It is 
+generally used to solve nonsymmetric, sparse matrices. However, as the lapack solver 
+converts the sparse matrix to a dense matrix, it will increase memory usage compared 
+to superlu.
 % Problem 7: Consequenses of solver choice, in regards to accuracy of probability conserved
 %            Add plot of deviation for both single- and double-slit
+Since we use a solver for sparse matrices, we decrease number of computations performed 
+compared to solver using dense matrix. We check if the total probability is conserved 
+over time, by plotting the deviation $s$ as
+\begin{align*}
+    s^{n} = 1 - \sum_{\ivec , \jvec} p_{\ivec , \jvec}^{n} = 1 - \sum_{\ivec , \jvec} u_{\ivec , \jvec}^{n*} u_{\ivec , \jvec}^{n} \ .
+\end{align*}
+The deviation as a function of time is plotted in Figure \ref{fig:deviation}.
+\begin{figure}
+    \centering
+    \includegraphics[width=\linewidth]{images/probability_deviation.pdf}
+    \caption{Deviation for $t \in [0, T]$ where $T=0.008$.}
+    \label{fig:deviation}
+\end{figure}
 
 \subsection{Time evolution}\label{ssec:time_evolution}
 % Problem 8: Colormap, include plot of both Re and Im for different time steps

python_scripts/colormap.py

@@ -4,6 +4,17 @@ import ast
 import seaborn as sns
 
 sns.set_theme()
+params = {
+    "font.family": "Serif",
+    "font.serif": "Roman",
+    "text.usetex": True,
+    "axes.titlesize": "large",
+    "axes.labelsize": "large",
+    "xtick.labelsize": "large",
+    "ytick.labelsize": "large",
+    "legend.fontsize": "medium",
+}
+plt.rcParams.update(params)
 
 def plot():
     with open("data/color_map.txt") as f:

python_scripts/detector.py

@@ -4,6 +4,17 @@ import ast
 import seaborn as sns
 
 sns.set_theme()
+params = {
+    "font.family": "Serif",
+    "font.serif": "Roman",
+    "text.usetex": True,
+    "axes.titlesize": "large",
+    "axes.labelsize": "large",
+    "xtick.labelsize": "large",
+    "ytick.labelsize": "large",
+    "legend.fontsize": "medium",
+}
+plt.rcParams.update(params)
 
 def plot():
     files = [

python_scripts/heat_map.py

@@ -4,6 +4,19 @@ import matplotlib
 from matplotlib.animation import FuncAnimation
 import ast
 
+import seaborn as sns 
+params = {
+    "font.family": "Serif",
+    "font.serif": "Roman",
+    "text.usetex": True,
+    "axes.titlesize": "large",
+    "axes.labelsize": "large",
+    "xtick.labelsize": "large",
+    "ytick.labelsize": "large",
+    "legend.fontsize": "medium",
+}
+plt.rcParams.update(params)
+
 wave_arr = []
 fig = plt.figure()
 ax = plt.gca()

python_scripts/normalization.py

@@ -4,6 +4,19 @@ import matplotlib
 from matplotlib.animation import FuncAnimation
 import ast
 
+import seaborn as sns 
+params = {
+    "font.family": "Serif",
+    "font.serif": "Roman",
+    "text.usetex": True,
+    "axes.titlesize": "large",
+    "axes.labelsize": "large",
+    "xtick.labelsize": "large",
+    "ytick.labelsize": "large",
+    "legend.fontsize": "medium",
+}
+plt.rcParams.update(params)
+
 def plot():
     with open("data/probability_deviation.txt") as f:
         lines = f.readlines();
@@ -16,6 +29,7 @@ def plot():
             arr_narrow.append(float(tmp[1]))
             arr_wide.append(float(tmp[2]))
 
+        
         plt.plot(x,arr_narrow)
         plt.plot(x,arr_wide)
         plt.savefig("latex/images/probability_deviation.pdf")

python_scripts/plot_v.py

@@ -4,6 +4,19 @@ import matplotlib
 from matplotlib.animation import FuncAnimation
 import ast
 
+import seaborn as sns 
+params = {
+    "font.family": "Serif",
+    "font.serif": "Roman",
+    "text.usetex": True,
+    "axes.titlesize": "large",
+    "axes.labelsize": "large",
+    "xtick.labelsize": "large",
+    "ytick.labelsize": "large",
+    "legend.fontsize": "medium",
+}
+plt.rcParams.update(params)
+
 
 def plot():
     with open("v.txt") as f: